Hyperbola: Definition, Equation, Properties, Examples, Applications

Hyperbolas are conic sections generated by a plane intersecting the bases of a double cone. Hyperbolas can also be viewed as the locus of all points with a common distance difference between two focal points. Each branch of a hyperbola has a focal point and a vertex.

Hyperbola examples can be seen in real life. When two stones are tossed into a pool of calm water simultaneously, ripples form in concentric circles. The hyperbola is a curve formed when these circles overlap in points. Shadows cast on a wall by a home lamp is in the shape of a hyperbola. A hyperbola is an idea behind solving trilateration problems which is the task of locating a point from the differences in its distances to given points. Let’s dive in to learn about hyperbola in detail.

What is Hyperbola?

A hyperbola is a conic section created by intersecting a right circular cone with a plane at an angle such that both halves of the cone are crossed in analytic geometry. This intersection yields two unbounded curves that are mirror reflections of one another.

In other words, A hyperbola is defined as the locus of all points in a plane whose absolute difference of distances from two fixed points on the plane remains constant.
The foci (singular focus) are the fixed points. The constant is the eccentricity of a hyperbola, and the fixed line is the directrix. Eccentricity is a property of the hyperbola that indicates its lengthening and is symbolised by the letter \(e.\)

Terms related to hyperbola are as follows:
1. The Transverse Axis is the line perpendicular to the directrix and passing through the focus.
2. The Vertices are the point on the hyperbola where its major axis intersects.
3. The Centre is the midpoint of vertices of the hyperbola.
4. The Conjugate axis is the straight line perpendicular to the transverse axis passing through the centre of the hyperbola.
5. The chord which passes through any of the two foci and is perpendicular to the transverse axis is known as the Latus Rectum.

General Equation of Hyperbola

The equation of a hyperbola in the standard form is given by:

\(\frac{{{x^2}}}{{{a^2}}} – \frac{{{y^2}}}{{{b^2}}} = 1\)

Where,
\({b^2} = {a^2}\left( {{e^2} – 1} \right)\)
\(e = \sqrt {1 + \frac{{{b^2}}}{{{a^2}}}} \)
Equation of transverse axis \( = x\) axis
Equation of conjugate axis \( = y\) axis
Centre\( = \left( {0,\,0} \right)\)

Similarly, the equation of hyperbola whose centre \(\left( {m,\,n} \right)\) in the standard form is given by \(\frac{{{{\left( {x – m} \right)}^2}}}{{{a^2}}} – \frac{{{{\left( {y – n} \right)}^2}}}{{{b^2}}} = 1,\)

On expanding the above equation, the general equation of a hyperbola looks like \(a{x^2} + 2\,hxy + b{y^2} + 2\,gx + 2\,fy + c = 0.\)
But the above expression will represent a hyperbola if \(\Delta \ne 0\) and \({h^2} > ab\)
Where,
\(\Delta = \left| {\begin{array}{*{20}{c}} a&h&g\\ h&b&f\\ g&f&c \end{array}} \right|\)

The equation of a conjugate hyperbola in the standard form is given by \(\frac{{{y^2}}}{{{b^2}}} – \frac{{{x^2}}}{{{a^2}}} = 1.\) The conjugate hyperbola is shown below:

The important parameters in the hyperbola are tabled below:

What are the Properties of Hyperbola?

Some of the important properties of a hyperbola are as follows:

1. There exist two focus, or foci, in every hyperbola. The difference in the distances between the two foci at each point on the hyperbola is a constant.
2. The directrix is a straight line that runs parallel to the hyperbola’s conjugate axis and connects both of the hyperbola’s foci.
3. The Transverse axis is always perpendicular to the directrix.
4. The foci and the vertices lie on the transverse axis.
5. At the vertices, the tangent line is always parallel to the directrix of a hyperbola.
6. The length of the latus rectum is \(\frac{{2\,{b^2}}}{a}\) for the hyperbola \(\frac{{{x^2}}}{{{a^2}}} – \frac{{{y^2}}}{{{b^2}}} = 1.\)
7. If the lengths of the transverse and conjugate axes are equal, a hyperbola is said to be rectangular or equilateral.

Applications of Hyperbola in Real-life

The real-life function of the hyperbola are as follows:

1. In many sundials, hyperbolas can be seen. The sun circles the celestial sphere every day, and its rays sketch out a cone of light when they strike the point on a sundial. A conic section is formed by the intersection of this cone with the ground’s horizontal plane. The angle between the ground plane and the sunlight cone varies depending on your location and the Earth’s axial tilt, which varies periodically. This conic section is a hyperbola in the majority of populated latitudes and times of the year.

2. Trilateration is a technique for locating an exact position by calculating the distances between two sites. A hyperbola is an idea behind solving trilateration problems which is the task of locating a point from the differences in its distances to given points or, equivalently, the difference in arrival times of synchronised signals between the point and the given points.

3. Any orbiting body’s path is known as the Kepler orbit. It can be applied to any size particle as long as the orbital trajectory is caused solely by gravity. This orbit can be any of the four conic sections depending on the orbital parameters, such as size and form (eccentricity). The path of such a particle is a hyperbola if the eccentricity e of the orbit is bigger than \(1.\)

4. For all nuclear cooling towers and several coal-fired power facilities, the hyperboloid is the design standard. It has a strong structural foundation and can be constructed with straight steel beams.

5. Hyperbolic shadows are cast on a wall by a home lamp.

6. The hyperbolic paraboloid geometry of Dulles Airport, created by Eero Saarinen, is unique. A hyperbolic paraboloid is a three-dimensional curve with a hyperbola in one cross-section and a parabola in the other.

7. In \(1953,\) a pilot flew faster than the speed of sound over an Air Force base. He wreaked havoc on the base’s infrastructure. A cone-like wave is created when an aircraft travels faster than the speed of sound. It’s a hyperbola when the cone meets the ground. Every point on the curve is hit by the sonic boom at the same time. Outside of the bend, no sound is heard. The “Sonic Boom Curve” is the name given to the hyperbola.

8. When two stones are tossed into a pool of calm water at the same time, ripples form in concentric circles. The hyperbola is a curve formed when these circles overlap in points.

Solved Examples – Hyperbola

Q.1. What will be the absolute difference of the focal distances of any point on the hyperbola \(9\,{x^2} – 16\,{y^2} = 144?\)
Ans: Given, \(9\,{x^2} – 16\,{y^2} = 144\)
\( \Rightarrow \frac{{{x^2}}}{{16}} – \frac{{{y^2}}}{9} = 1\)
Here \(a = 4\) and \(b = 3\)
The absolute difference of the distances of any point from their foci on a hyperbola is constant, which is the length of the transverse axis.
i.e. the absolute difference of the focal distances of any point on a hyperbola \( = 2\,a = 8.\)

Q.2. What will the eccentricity of hyperbola \(16\,{x^2} – 25\,{y^2} = 400?\)
Ans: Given, \(16\,{x^2} – 25\,{y^2} = 400\)
\( \Rightarrow \frac{{{x^2}}}{{25}} – \frac{{{y^2}}}{{16}} = 1\)
Here, \(a = 5\) and \(b = 4\)
So, \(e = \sqrt {1 + \frac{{{b^2}}}{{{a^2}}}} = \sqrt {1 + \frac{{16}}{{25}}} = \frac{{\sqrt {41} }}{5}\)

Q.3. What will the coordinate of foci of hyperbola \(16\,{x^2} – 25\,{y^2} = 400?\)
Ans: Given, \(16\,{x^2} – 25\,{y^2} = 400\)
\( \Rightarrow \frac{{{x^2}}}{{25}} – \frac{{{y^2}}}{{16}} = 1\)
Here, \(a = 5\) and \(b = 4\)
So, \(e = \sqrt {1 + \frac{{{b^2}}}{{{a^2}}}} = \sqrt {1 + \frac{{16}}{{25}}} = \frac{{\sqrt {41} }}{5}\)
So, coordinate of foci \( = \left( { \pm ae,\,o} \right) = \left( { \pm \sqrt {41} ,\,0} \right)\)

Q.4. Find the length of the latus rectum of hyperbola \(9\,{x^2} – 16\,{y^2} = 144?\)
Ans: Given, \(9\,{x^2} – 16\,{y^2} = 144\)
\( \Rightarrow \frac{{{x^2}}}{{16}} – \frac{{{y^2}}}{{9}} = 1\)
Here \(a = 4\) and \(b = 3\)
Hence, the length of the latus rectum of hyperbola \( = \frac{{2\,{b^2}}}{a} = \frac{{2 \times 9}}{4} = \frac{9}{2}.\)

Q.5. If the length of the transverse axis and conjugate axis of a hyperbola is \(10\) and \(8\) respectively, then find the eccentricity of that hyperbola?
Ans: Since the length of the transverse axis and conjugate axis of a hyperbola is \(10\) and \(8,\) respectively.
So, \(2\,a = 10,\,2\,b = 8\)
\(a = 5,\,b = 4\)
So, \(e = \sqrt {1 + \frac{{{b^2}}}{{{a^2}}}} = \sqrt {1 + \frac{{16}}{{25}}} = \frac{{\sqrt {41} }}{5}\)

Summary

In this article, we have learnt about hyperbola, equations, their properties and their applications in the real world. A hyperbola is the locus of all points in a plane whose absolute difference of distances from two fixed points on the plane remains constant. It can be seen in many sundials, solving trilateration problems, home lamps, etc. We have seen its immense uses in the real world, which is also significant role in the mathematical world.

FAQs

Q.1. What is the difference between parabola and hyperbola?
Ans: A parabola is a locus that contains all points with the same distance from a focus and a directrix. On the other hand, a hyperbola is a locus of all the points where the distance between two foci is constant. A hyperbola is an open curve with two branches and two foci and directrices, whereas a parabola is an open curve with one focus and directrix. A parabola’s eccentricity is one, whereas a hyperbola’s eccentricity is larger than one. When a plane intersects a cone at its slant height, a parabola is generated. On the other hand, a hyperbola is generated when a plane hits a cone at its perpendicular height.

Q.2.What is meant by asymptotes in hyperbola?
Ans: Asymptotes in hyperbola are the straight lines, tangent to the hyperbola where the point of contact tends to infinity.

Q.3. What is the focus of a hyperbola?
Ans: A hyperbola’s foci are the two fixed points that are located inside each curve of the hyperbola. For the standard hyperbola \(\frac{{{x^2}}}{{{a^2}}} – \frac{{{y^2}}}{{{b^2}}} = 1,\) the coordinate of foci are \(\left( { \pm ae,\,0} \right)\) where \(e = \sqrt {1 + \frac{{{b^2}}}{{{a^2}}}} \)

Q.4. What is the hyperbola curve?
Ans: A hyperbola is a two-branched open curve formed by intersecting a plane with both halves of a double cone. The plane need not be parallel to the cone’s axis; the hyperbola will be symmetrical regardless.

Q.5. What is the formula of the eccentricity of a hyperbola?
Ans: The eccentricity of a hyperbola \(\frac{{{x^2}}}{{{a^2}}} – \frac{{{y^2}}}{{{b^2}}} = 1\) is given by \(e = \sqrt {1 + \frac{{{b^2}}}{{{a^2}}}} \)

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